  RUS  ENG ЖУРНАЛЫ   ПЕРСОНАЛИИ   ОРГАНИЗАЦИИ   КОНФЕРЕНЦИИ   СЕМИНАРЫ   ВИДЕОТЕКА   ПАКЕТ AMSBIB  Ближайшие семинары Календарь семинаров Список семинаров Архив по годам Регистрация семинара Поиск RSS Ближайшие семинары

Семинар по геометрической топологии
26 октября 2017 г. 14:00–16:00, г. Москва, Матфак ВШЭ (ул. Усачёва, 6), ауд. 209  Projected embeddings

С. А. Мелихов

 Количество просмотров: Эта страница: 71

Аннотация: An obvious necessary condition for a map $f: N\to M$ to lift to an embedding $N\to M\times\mathbb R^k$ is the existence of a $\mathbb Z_2$-equivariant map $\Delta_f\to S^{k-1}$, where $\Delta_f$ is the set of pairs $(x,y)\in N\times N$ such that $f(x)=f(y)$ and $x\ne y$. This condition is obviously not sufficient for the degree 3 covering $f: S^1\to S^1$ (with $k=1$), but M. Skopenkov proved that it is sufficient for maps $f$ of a trivalent graph into $\mathbb R^1$ (with $k=1$). Also, Haefliger proved in 1963 that it is sufficient in the case where $M$ is a point, $N$ is a smooth manifold and $2k\ge 3(\dim N+1)$.
We prove that the condition is sufficient when $f$ is a generic PL map or a generic smooth map, $n\le m$, $2(m+k)\ge 3(n+1)$ and $4n-3m\le k$, where $n=\dim N$, $m=\dim M$. In both cases the constructed lift will be only a piecewise-smooth embedding. When $f$ is a generic PL map, we can find a lift that is a PL embedding by some additional work. But if $f$ is a generic smooth map and we want the lift to be a smooth embedding, we must assume additionally that either $3n-2m\le k$ or $f$ has no singularities of type $\Sigma^{1,1}$.
One could try to prove these (or similar) results by some version of Haefliger's generalization of the Whitney trick. But, unfortunately, it does not work. We use a new kind of "Whitney trick", which in contrast to Haefliger's is described by an explicit formula.

 ОТПРАВИТЬ:       Обратная связь: math-net2019_12 [at] mi-ras ru Пользовательское соглашение Регистрация Логотипы © Математический институт им. В. А. Стеклова РАН, 2019